Numerical Evidence for Many-Body Localization in Two and Three Dimensions.

Disorder and interactions can lead to the breakdown of statistical mechanics in certain quantum systems, a phenomenon known as many-body localization (MBL). Much of the phenomenology of MBL emerges from the existence of l bits, a set of conserved quantities that are quasilocal and binary (i.e., possess only ±1 eigenvalues). While MBL and l bits are known to exist in one-dimensional systems, their existence in dimensions greater than one is a key open question. To tackle this question, we develop an algorithm that can find approximate binary l bits in arbitrary dimensions by adaptively generating a basis of operators in which to represent the l bit. We use the algorithm to study four models: the one-, two-, and three-dimensional disordered Heisenberg models and the two-dimensional disordered hard-core Bose-Hubbard model. For all four of the models studied, our algorithm finds high-quality l bits at large disorder strength and rapid qualitative changes in the distributions of l bits in particular ranges of disorder strengths, suggesting the existence of MBL transitions. These transitions in the one-dimensional Heisenberg model and two-dimensional Bose-Hubbard model coincide well with past estimates of the critical disorder strengths in these models, which further validates the evidence of MBL phenomenology in the other two- and three-dimensional models we examine. In addition to finding MBL behavior in higher dimensions, our algorithm can be used to probe MBL in various geometries and dimensionality.